1 the Tangent Bundle and Vector Bundle

Home , Bundle (mathematics), Cotangent bundle, Cotangent space, Dual bundle, Tangent bundle, Vector bundle

Tangent bundle, vector bundles and vector ﬁelds

by Min Ru

The aim of this section is to introduce the tangent bundle TX for a diﬀerential manifold X. Intuitively this is the object we get by gluing at each point p ∈ X the corresponding

tangent space TpX. The differentiable structure on X induces a differentiable structure on TX making it into a differentiable manifold of dimension 2 dim(X). The tangent bundle TX is the most important example of what is called a vector bundle over X(see the definition below).

1.1. Review of the tangent space TpX: Let X be a smooth diﬀerential manifold of dimension m and let p ∈ X. The tangent

space TpX is a collection of tangent vectors vp to X at the point p. A tangent vector vp ∞ is a map vp : C (X) → R such that (i) vp(af + bg) = avp(f)+ bvp(g), (ii) vp(fg) = m m f(p)vp(g)+ g(p)vp(f). Let (U, φ) a local coordinate for X at p, let R = Ru1,...,um and write ∂ φ = (x1,...,xm). Then we have special tangent vectors { | , 1 ≤ k ≤ m} (called the ∂xk p partial derivatives) ∂ | : C∞(X) → R ∂xk p deﬁned by ∂ ∂(f ◦ φ−1) | (f)= | . ∂xk p ∂uk φ(p ∂ Then { | , 1 ≤ k ≤ m} forms a basis for T X, moreover(from the proof), for every v ∈ ∂xk p p p TpX, we can write m ∂ v = v (xk) | . p p ∂xk p kX=1 1.2. Construction of the tangent bundle TX:

1 Let X be a smooth diﬀerential manifold of dimension m. Let

TX = ∪p∈X TpX = {(p, v) | p ∈ X, v ∈ TpX}.

Let π : TX → X be the natural projection map with π :(p, v) 7→ p. For a given point p ∈ X

−1 the fiber π ({p}) of π is the m-dimensional tangent space TpX at p. The triple (TX,X,π) is called the tangent bundle of X. We can put a differentiable structure on TX making it into a differentiable manifold of dimension 2 dim(X) as follows:

Let X be a diﬀerential manifold with maximal atlas A. Let x : U → Rm in A be a chart for X and deﬁne −1 m m ψ˜U : π (U) → R × R

by m ∂ ψ˜ : (p, a | ) 7→ (x(p), (a ,...,a )). U k ∂xk p 1 m kX=1 −1 m m Then it is easy to check that ψ˜U is one-to one and ψ˜U (π (U)) is an open set in R × R . We now check that overlap(transition) maps are smooth maps. In fact, Let (U, x) and (V,y) be two charts in A such that p ∈ U ∩ V . Then the overlap(transition) map

−1 −1 m m ψ˜V ◦ (ψ˜U ) : ψ˜U (π (U ∩ V )) → R × R

is given by

m m −1 ∂y1 ∂ym (a, b) 7→ (y ◦ x (a), | −1 b ,..., | −1 b ). ∂xk x (a) k ∂xk x (a) k kX=1 kX=1 −1 −1 Since X is a smooth manifold, y ◦ x is smooth, hence ψ˜V ◦ (ψ˜U ) is also smooth. Let

∗ −1 A = {(π (U), ψ˜U ) | (U, x) ∈ A},

then A∗ is a C∞ atlas. So TX is an 2m smooth manifold. It is trivial that the projection map π : TX → X is also smooth.

2 1.3. The tangent bundle, cotangent bundle and the deﬁnition of general vector bundle.

−1 For each point p ∈ X the fiber π ({p}) is the tangent space TpX of X at p hence an m- m −1 m dimensional vector space. For a chart x : U → R is A, we define ψU : π (U) → U × R by m ∂ ψ : (p, a | ) 7→ (p, (a ,...,a )). U k ∂xk p 1 m kX=1 Obviously ψU is a diffeomorphism. Further more, it has the following important property: m the restriction of ψU to the tangent space TpX, i.e. ψp = ψU |TpX : TpX →{p} × R is given by m ∂ ψ : a | 7→ (a ,...,a ), p k ∂xk p 1 m kX=1 −1 m so it is a vector space isomorphism. The map ψU : π (U) → U ×R is called a bundle chart.

In summary: For a smooth manifold X, we get a triple (TX,X,π), which is called the tangent bundle of X, where π is a continuous surjective map(natural projection), TX is a smooth diﬀerential manifold of dimension 2 dim(X). Further, it satisﬁes the following property:

−1 (i) For each p ∈ X, the ﬁber π ({p})= Tp(X) is an m-dimensional vector space.

−1 (ii) For each p ∈ X there exists a bundle chart (π (U), ψU ) (some book called it

−1 m trivialization) such that ψU : π (U) → U × R is a smooth diﬀeomorphism and for all m q ∈ U, the map ψq = ψU |Tq(X) : Tq(X) →{q} × R is a vector space isomorphism.

A smooth map v : X → TX is called a smooth vector ﬁeld (or smooth section) if π ◦ v(p)= p for each p ∈ X.

Finally, motivated by the above construction, we introduce the following general deﬁni- tion:

3 Deﬁnition 1: Let E and X be smooth manifolds and π : E → X be a smooth surjective map. The triple (E,X,π) is called a (smooth) vector bundle of rank k over X if

−1 (i) For each p ∈ X, the ﬁber Ep = π ({p}) is a k-dimensional vector space.

−1 (ii) For each p ∈ X there exists a bundle chart (π (U), ψU ) (some book called it −1 k trivialization) such that ψU : π (U) → U × R is a smooth diﬀeomorphism and for all k q ∈ U, the map ψq = ψU |Eq : Eq(X) →{q} × R is a vector space isomorphism.

Vector bundles of rank 1 is also called the line bundle.

The vector bundle of rank r over X is said to be trivial if there exists a global bundle chart ψ : E → X × Rk.

Deﬁnition 2: Let (E,X,π) be a vector bundle over X. A smooth map σ : X → E is said to be a smooth section of the bundle (E,X,π) if π ◦ σ(p)= p for every p ∈ X. The set of all smooth sections is denoted by Γ(X, E) or just Γ(E).

Deﬁnition 3: Let (E,X,π) be a vector bundle of rank k over X. Let {Uα} be an open

−1 k covering of X and let ψα : π (Uα) → Uα × R be the trivialization. Then, on Uα ∩ Uβ =6 ∅, the composition map

−1 k k ψα ◦ ψβ :(Uα ∩ Uβ) × R → (Uα ∩ Uβ) × R

k is of the form, for every p ∈ Uα ∩ Uβ and b ∈ R ,

−1 ψα ◦ ψβ (p, b)=(p, gαβ(p)(b))

for some smooth map gαβ : Uα ∩ Uβ → GL(k, R) where GL(k, R) is the set of k × k non- singular matrices. The smooth GL(k, R)-valued maps {gαβ} are called the transition functions for a vector bundle E.

4 Examples

1. Let E = X × Rk. Then E is a vector bundle of rank k. In this case, the trivialization map is an identity map. This bundle is called the trivial bundle.

2. Let E = TX = ∪p∈X Tp(X). It is called the tangent bundle, denoted by TX. The rank of this bundle is m (the dimension of TX as a manifold is 2m), where dim X = m. Let

1 m (U, φU ) be a chart of X with coordinate functions x ,...,x . Then it deﬁnes a trivialization −1 m ψU : π (U) → U × R by

m ∂ ψ : (p, a | ) 7→ (p, (a ,...,a )). U k ∂xk p 1 m kX=1

We now calculate the transition functions. Let(U, φU ), (V,φV ) two charts on X, with coordinate functions x1,...,xm and y1,...,ym respectively, where U ∩ V =6 ∅. For every

m b =(b1,...,bm) ∈ R , and p ∈ U ∩ V ,

m −1 ∂ ψV (p, b)=(p, bi i |p). i ∂y X=1 Since, on U ∩ V , ∂ m ∂xj ∂ i |p= i j |p, ∂y j ∂y ∂x X=1 we conclude that, on U ∩ V ,

m m m j −1 ∂ ∂x ∂ ψV (p, b)=(p, bi i |p)=(p, bi i j |p). i ∂y j i ∂y ∂x X=1 X=1 X=1 Hence, m 1 m m −1 ∂x ∂x ψU ◦ ψV (p, b)=(p, bi i ,..., bi i ). i ∂y i ∂y X=1 X=1 This means the transition map gUV is, for every p ∈ U ∩ V ,

∂xi gUV (p)= j |φV (p). ∂y !1≤i,j≤m

5 3. Besides the tangent bundle TX above, we also have the cotangent bundle T ∗X as follows:

Consider a smooth manifold X of dimension m. The dual space to the tangent space TpX, ∗ m p ∈ X, is called the cotangent space to X at p, denoted by Tp X. Suppose that x : U → R ∂ be a local coordinates for X at p, then { | , 1 ≤ k ≤ m} forms a basis for T X, i.e. ∂xk p p ∂ for every v ∈ T X. The dual basis to { | , 1 ≤ k ≤ m} is traditionally denoted by p p ∂xk p k {dx |p, 1 ≤ k ≤ m}. (Sometimes, I may drop the subscript p from the notation.) Thus an m ∗ k arbitrary element of Tp X is expressed as akdx |p for some ak ∈ R. The disjoint union k=1 of all the cotangent space X ∗ ∗ T X = ∪p∈X Tp X

is called the cotangent bundle of X. The cotangent bundle can be given a smooth structure making it into a manifold of dimension 2 dim(X) by an argument very similar to the one for the tangent bundle as above). It is easy to verify that the transition functions for T ∗X is ∂yi gUV (p)= j |φU (p). ∂x !1≤i,j≤m

4. Tensor Bundles. Consider the (r, s)−type tensor space

r ∗ ∗ Ts (p)= Tp(x) ⊗ Tp(X) ⊗ ⊗Tp (X) ⊗···⊗ Tp(X) ,

where the ﬁrst products for V is taken r times, and the second products for V ∗ is taken s times. Let r r Ts = ∪p∈X Ts (p).

r Then, similar above, we can show that Ts is a vector bundle, which is called an (r, s)-type tensor bundle on X.

5. Bundle Operations. Let (E,X,π) be a vector bundle, we can deﬁne, in an obvious way, the dual bundle (E∗,X,π), which is called the dual bundle. Similarly, let E, E′ be two vector bundles, we can deﬁne E ⊕ E′ and E ⊗ E′ the direct sum and the the tensor product bundles.

6 2 Smooth Vector ﬁelds

Let X be a smooth manifold of dimension m. A vector ﬁeld v on X is a section of the tangent bundle TX, ie v : X → TX such that π ◦ v(p)= p for every p ∈ X. In other words. A vector ﬁeld on X is a map v which assigns to each point p ∈ X a tangent vector

v(p)= vp ∈ Tp(X).

Let x : U → Rm be a local chart of X, and p ∈ U, then

m i ∂ v(p)= vp(x ) i |p . i ∂x X=1 i i i The real-valued functions v : U → R deﬁned by v (p) = vp(x ), 1 ≤ i ≤ m, are called the components of v related to the local chart x : U → Rm.

A vector ﬁeld is said to be smooth if v is a smooth section tangent bundle TX, i.e. v is smooth as a map. It can be checked (we omit it here) that v is smooth if and only if its components are smooth for all charts in some atlas for X. Denote by Γ(X) the set of all smooth vector ﬁelds on X. We have the following algebra structure on Γ(X):

For v,w ∈ Γ(X), a ∈ R and f ∈ C∞(X):

(i) v + w ∈ Γ(X), i.e. (v + w)(p)= v(p)+ w(p);

(ii) av ∈ Γ(X), i.e. (av)(p)= av(p);

(iii) fv ∈ Γ(X), i.e. (fv)(p)= f(p)v(p).

There is another way of thinking about vector ﬁelds: Recall that, for every p ∈ X, the ∞ tangent vector vp map vp : C (X) → R such that (i) vp(af + bg) = avp(f)+ bvp(g), (ii)

vp(fg)= f(p)vp(g)+ g(p)vp(f). Since v assigns at each point p ∈ X a tangent vector vp, we ∞ can deﬁne, for each f ∈ C (X), v(f) as a function on X by v(f)(p)= vp(f). If the vector

7 ﬁeld is smooth, then v(f) is also a smooth function on X for every f ∈ C∞(X). Hence, we can think of a smooth vector ﬁeld v ∈ Γ(X) as a map

v : C∞(X) → C∞(X) by f 7→ v(f), where (vf)(p)= vp(f).

The smooth vector ﬁeld v : C∞(X) → C∞(X) also satisﬁes the following properties: For every f,g ∈ C∞(X),

(i) v(af + bg)= av(f)+ bv(g),

(ii) v(fg)= fv(g)+ gv(f).

An exterior diﬀerential p-form is a section of p T ∗(X). Inalocalchart (U, (x1,...,xm)), an exterior diﬀerential p-form V

j1 jp ω = aj1···

Λp(X) will denote the vector space of exterior diﬀerential p-forms. Γ(X) will denote the vector space of (smooth) vector ﬁelds.

s Tr (X) denote the vector space of (r, s)-tensor ﬁelds.

8 3 Lie bracket of smooth vector ﬁelds

Let v,w be two smooth vector ﬁelds (thought as maps from C∞(X) → C∞(X)). Deﬁne the Lie bracket of v and w, denoted by

[v,w] : C∞(X) → C∞(X),

by [v,w](f)= v(w(f)) − w(v(f)).

The Lie bracket has the following properties:

(i) (R-linearity): [au + bw, v]= a[u, v]+ b[w, v];

(ii) (skew-symmetry): [v,w]= −[w, v];

(iii) (Jacobi-identity) [u, [v,w]] + [w, [u, v]] + [v, [w,u]] = 0;

(iv) for every f,g ∈ C∞(X), [fv,w]= f[v,w] − (w(f))v, [v,gw]= g[v,w]+(v(g))w;

m i ∂ m i ∂ (v) In local coordinates, if v = i=1 v ∂xi , and w = i=1 w ∂xi , then

P m i Pi j ∂w j ∂v ∂ [v,w]= v j − w j i . i,j ∂x ∂x ! ∂x X=1

4 Exterior Diﬀerentials

Recall that an exterior diﬀerential p-form is a section of p T ∗(X). Inalocalchart (U, (x1,...,xm)), an exterior diﬀerential p-form V

j1 jp ω = aj1···

9 Let m Λ(X)= Λp(X). p M=0 It is called the algebra of exterior diﬀerential forms. Note that it has the structure of an algebra with respect to wedge product “∧”.

We now deﬁne the exterior diﬀerential operator d : Λp(X) → Λp+1(X) as follows, for p every ω ∈ Λ (X), and for every X1,...,Xp+1 ∈ Γ(X),

p+1 dω(X1,...,Xp+1) = Xi(ω(X1,..., Xˆi,...,Xp+1) i X=1 i+j + (−1) ω([Xi,Xj],X1,..., Xˆi,..., Xˆj,...,Xp+1). i

j1 jp ω = aj1···

j1 jp dω = daj1···

d(η ∧ ξ)= dη ∧ ξ +(−1)pη ∧ dξ.

According to the deﬁnition, for f ∈ Λ0(X), we have d(f)(X)= X(f), and for ω ∈ Λ1(X), we have dω(X,Y )= X(ω(Y )) − Y (ω(X)) − ω([X,Y ]).

Now let F : M → N be a smooth map, then we have d(F ∗ω)= F ∗(dω) for any diﬀerential form ω on N.

Note that we can deﬁne the integration of a k-form with compact support on a k- dimensional (orientable) manifold, and have the following statement of Stokes’ formula: Let

10 M be a smooth compact manifold of dimension n with boundary, and ω is a (n − 1)-form on M. Then dω = ω. ZM Z∂M